Hyper-Wiener index of zero-divisor graphs of matrices: A computational study

Sahil Sharma; Omaima Al Shanqiti; Vijay Kumar Bhat; Sahil Sharma; Omaima Al Shanqiti; Vijay Kumar Bhat

doi:10.3934/math.20261397

AIMS Mathematics

2026, Volume 11, Issue 11: 36080-36099. doi: 10.3934/math.20261397

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Hyper-Wiener index of zero-divisor graphs of matrices: A computational study

1.
School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, Jammu and Kashmir, India
2.
Department of Mathematics, Umm Al-Qura University, Makkah 24381, Saudi Arabia

Received: 20 August 2026 Revised: 03 October 2026 Accepted: 15 October 2026 Published: 03 November 2026
MSC : 05C10, 05C12, 05C40, 05C85, 05C92, 11C20

Topological indices, which are numerical values derived from the structure (topology) of a graph, were used to characterize its attributes. With applications in a variety of fields including network research, chemistry, and cryptography, the Hyper-Wiener index is a crucial topological descriptor for examining the structural characteristics of graphs. In this work, the Hyper-Wiener index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non-commutative ring was computed. In this study, we computed the Hyper-Wiener index of the directed and undirected zero-divisor graphs obtained from upper triangular matrix $ (M_3) $.
- upper triangular matrix,
- zero-divisor graphs,
- topological indices,
- Hyper-Wiener index
Citation: Sahil Sharma, Omaima Al Shanqiti, Vijay Kumar Bhat. Hyper-Wiener index of zero-divisor graphs of matrices: A computational study[J]. AIMS Mathematics, 2026, 11(11): 36080-36099. doi: 10.3934/math.20261397

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Abstract

Topological indices, which are numerical values derived from the structure (topology) of a graph, were used to characterize its attributes. With applications in a variety of fields including network research, chemistry, and cryptography, the Hyper-Wiener index is a crucial topological descriptor for examining the structural characteristics of graphs. In this work, the Hyper-Wiener index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non-commutative ring was computed. In this study, we computed the Hyper-Wiener index of the directed and undirected zero-divisor graphs obtained from upper triangular matrix $ (M_3) $.

References

[1]	I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208–226. https://doi.org/10.1016/0021-8693(88)90202-5 doi: 10.1016/0021-8693(88)90202-5
[2]	D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447. https://doi.org/10.1006/jabr.1998.7840 doi: 10.1006/jabr.1998.7840
[3]	S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings, 1 (2002), 203–211.
[4]	F. R. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65 (2002), 206–214. https://doi.org/10.1007/s002330010128 doi: 10.1007/s002330010128
[5]	M. Behboodi, Zero-divisor graphs for modules over commutative rings, J. Commut. Algebra, 4 (2012), 175–197.
[6]	R. Xing, B. Zhou, X. Qi, Hyper-Wiener index of unicyclic graphs, MATCH Commun. Math. Comput. Chem., 66 (2011), 315–328.
[7]	C. Ouyang, L. Qi, X. Yuan, The first few unicyclic and bicyclic hypergraphs with largest spectral radii, Linear Algebra Appl., 527 (2017), 141–162. https://doi.org/10.1016/j.laa.2017.04.008 doi: 10.1016/j.laa.2017.04.008
[8]	L. Feng, W. Liu, K. Xu, The Hyper-Wiener index of bicyclic graphs, Utilitas Math., 84 (2011), 97–104.
[9]	E. A. Castro, I. Gutman, D. J. G. Marino, P. J. Peruzzo, Upgrading the Wiener index, J. Serbian Chem. Soc., 67 (2002), 647–651.
[10]	M. Knor, R. Škrekovski, A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Contempor., 11 (2016), 327–352.
[11]	I. Gutman, W. Linert, I. Lukovits, A. A. Dobrynin, Trees with external Hyper-Wiener index: Mathematical basis and chemical applications, J. Chem. Inf. Comput. Sci., 37 (1997), 349–354. https://doi.org/10.1021/ci960139m doi: 10.1021/ci960139m
[12]	I. Gutman, B. Furtula, Hyper-Wiener index versus Wiener index: Two highly correlated structure descriptors, Monatsh. Chem., 134 (2003), 975–981. https://doi.org/10.1007/s00706-003-0003-7 doi: 10.1007/s00706-003-0003-7
[13]	M. F. Nadeem, S. Zafar, Z. Zahid, On topological properties of the line graphs of subdivision graphs of certain nanostructures, Appl. Math. Comput., 273 (2016), 125–130. https://doi.org/10.1016/j.amc.2015.10.010 doi: 10.1016/j.amc.2015.10.010
[14]	H. Q. Liu, M. Lu, A unified approach to extremal cacti for different indices, MATCH Commun. Math. Comput. Chem., 58 (2007), 183–194.
[15]	A. Vijayabarathi, G. S. G. N. Anjaneyulu, Wiener index of a graph and chemical applications, Int. J. ChemTech Res., 5 (2013), 1847–1853.
[16]	A. El-Mesady, Y. S. Hamed, K. M. Abualnaja, A novel application on mutually orthogonal graph squares and graph-orthogonal arrays, AIMS Math., 7 (2022), 7349–7373. https://doi.org/10.3934/math.2022410 doi: 10.3934/math.2022410
[17]	A. El-Mesady, O. Bazighifan, H. M. Shabana, On graph-transversal designs and graph-authentication codes based on mutually orthogonal graph squares, J. Math., 2022 (2022), 8992934. https://doi.org/10.1155/2022/8992934 doi: 10.1155/2022/8992934
[18]	M. R. Ahmadi, R. Jahani-Nezhad, Wiener index and energy of the graph of $\Gamma(\mathbb{Z}_n)$, Iranian J. Math. Chem., 2 (2011), 45–51.
[19]	S. Suthar, Wiener index and energy of the line graph of the $\Gamma(\mathbb{Z}_n)$ zero-divisor graph for $n = pq$ and $n = p^2$, arXiv preprint, arXiv: 1806.08944, 2018.
[20]	S. Chattopadhyay, K. L. Patra, B. K. Sahoo, Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_n$'s zero-divisor graph, Linear Algebra Appl., 584 (2020), 267–286. https://doi.org/10.1016/j.laa.2019.08.015 doi: 10.1016/j.laa.2019.08.015
[21]	P. Singh, V. K. Bhat, Graph invariants of the line graph of zero divisor graph of $ \mathbb {Z}_{n} $, J. Appl. Math. Comput., 68 (2022), 1271–1287. https://doi.org/10.1007/s12190-021-01567-0 doi: 10.1007/s12190-021-01567-0
[22]	M. I. Bhat, S. Pirzada, On strong metric dimension of zero-divisor graphs of rings, Korean J. Math., 27 (2019), 563–580. https://doi.org/10.11568/kjm.2019.27.3.563 doi: 10.11568/kjm.2019.27.3.563
[23]	S. Pirzada, R. Raja, On the metric dimension of a zero-divisor graph, Commun. Algebra, 45 (2017), 1399–1408. https://doi.org/10.1080/00927872.2016.1175602 doi: 10.1080/00927872.2016.1175602
[24]	V. K. Bhat, P. Singh, On zero-divisor graph of matrix ring $M_n(\mathbf{Z}_p)$, Int. J. Appl. Math., 34 (2021), 1111–1122. https://doi.org/10.12732/ijam.v34i6.5 doi: 10.12732/ijam.v34i6.5
[25]	S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274 (2004), 847–855. https://doi.org/10.1016/S0021-8693(03)00435-6 doi: 10.1016/S0021-8693(03)00435-6

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